Advanced Calculus Single Variable

2.10 Completeness of ℝ

By Theorem 2.7.9, between any two real numbers, points on the number line, there exists a
rational number. This suggests there are a lot of rational numbers, but it is not clear
from this Theorem whether the entire real line consists of only rational numbers.
Some people might wish this were the case because then each real number could be
described, not just as a point on a line but also algebraically, as the quotient of integers.
Before 500 B.C., a group of mathematicians, led by Pythagoras believed in this,
but they discovered their beliefs were false. It happened roughly like this. They
knew they could construct the square root of two as the diagonal of a right triangle
in which the two sides have unit length; thus they could regard

√2

as a number.
Unfortunately, they were also able to show

√2

could not be written as the quotient of
two integers. This discovery that the rational numbers could not even account for
the results of geometric constructions was very upsetting to the Pythagoreans,
especially when it became clear there were an endless supply of such “irrational”
numbers.

This shows that if it is desired to consider all points on the number line, it is necessary to
abandon the attempt to describe arbitrary real numbers in a purely algebraic manner using
only the integers. Some might desire to throw out all the irrational numbers, and considering
only the rational numbers, confine their attention to algebra, but this is not the approach to
be followed here because it will effectively eliminate every major theorem of calculus. In
this book real numbers will continue to be the points on the number line, a line
which has no holes. This lack of holes is more precisely described in the following
way.

s for all s ∈ S. If S is a nonempty set in ℝ which isbounded above, then a number, l which has the property that l is an upper bound andthat every other upper bound is no smaller than l is called a least upper bound, l.u.b.

(S)

or often sup

(S)

. If S is a nonempty set bounded below, define the greatest lower bound,g.l.b.

(S )

or inf

(S )

similarly. Thus g is the g.l.b.

(S)

means g is a lower bound for Sand it is the largest of all lower bounds. If S is a nonempty subset of ℝ which is notbounded above, this information is expressed by saying sup

(S)

= +∞ and if S is notbounded below, inf

(S)

= −∞.

Every existence theorem in calculus depends on some form of the completeness axiom. In
an appendix, there is a proof that the real numers can be obtained as equivalence classes of
Cauchy sequences of rational numbers.

Axiom 2.10.2(completeness) Every nonempty set of real numbers which is boundedabove has a least upper bound and every nonempty set of real numbers which is boundedbelow has a greatest lower bound.

It is this axiom which distinguishes Calculus from Algebra. A fundamental result about
sup and inf is the following.

Proposition 2.10.3Let S be a nonempty set and suppose sup

(S)

exists. Then for everyδ > 0,

S ∩ (sup(S)− δ,sup(S)] ⁄= ∅.

If inf

(S)

exists, then for every δ > 0,

S ∩[inf (S ),inf(S)+ δ) ⁄= ∅.

Proof:Consider the first claim. If the indicated set equals ∅, then sup

(S )

−δ is an upper
bound for S which is smaller than sup

(S )

, contrary to the definition of sup

(S)

as the least
upper bound. In the second claim, if the indicated set equals ∅, then inf